Long Answer Questions 2 — Question 2

2x + 3 = 0

⇒ 2x = -3

⇒ x = $-\dfrac{3}{2}$

Substituting value of x in equation 2x² + 11x + 12, we get :

$\Rightarrow 2 \times \Big(-\dfrac{3}{2}\Big)^2 + 11 \times \Big(-\dfrac{3}{2}\Big) + 12 \\[1em] \Rightarrow 2 \times \dfrac{9}{4} - \dfrac{33}{2} + 12 \\[1em] \Rightarrow \dfrac{9}{2} - \dfrac{33}{2} + 12 \\[1em] \Rightarrow \dfrac{9 - 33 + 24}{2} \\[1em] \Rightarrow \dfrac{0}{2} \\[1em] \Rightarrow 0.$

Since, remainder = 0.

∴ 2x + 3 is a factor of the polynomial 2x² + 11x + 12.

Solving polynomial, 2x² + 11x + 12, we get :

⇒ 2x² + 8x + 3x + 12

⇒ 2x(x + 4) + 3(x + 4)

⇒ (2x + 3)(x + 4).

Solving polynomial, x² + 3x - 4, we get :

⇒ x² + 4x - x - 4

⇒ x(x + 4) - 1(x + 4)

⇒ (x - 1)(x + 4).

∴ (x - 1) and (x + 4) are factors of x² + 3x - 4.

∴ On multiplying polynomial, 2x² + 11x + 12 by (x - 1) it will be divisible by x² + 3x - 4.

⇒ (2x² + 11x + 12)(x - 1)

⇒ 2x³ - 2x² + 11x² - 11x + 12x - 12

⇒ 2x³ + 9x² + x - 12.

Hence, the resulting polynomial = 2x³ + 9x² + x - 12.